Theorem rexralbidv

Description: Formula-building rule for restricted quantifiers (deduction form). (Contributed by NM, 28-Jan-2006)

	Ref	Expression
Hypothesis	2rexbidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	rexralbidv	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		2rexbidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	ralbidv	⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐵 𝜓 ↔ ∀ 𝑦 ∈ 𝐵 𝜒 ) )
3	2	rexbidv	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 ) )